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openings design in underground mines, different approaches, kirscha formulae for circular opening, plastic xzone effect on stability of opening, radial and tangential stresses distribution

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- 1. DESIGN OF OPENINGS U.Siva Sankar Sr. Under Manager Project Planning Singareni Collieries Company Ltd E-Mail :ulimella@gmail.com or uss_7@yahoo.com Visit at:www.slideshare.net/sankarsulimella THE MINING ENVIRONMENT IN-SITU STRESSES 1
- 2. Stress beneath surface Y σy τxy σx r y θ X xNatural stresses which exist prior to excavation. The natural stress field can becomposed of gravitational stress, tectonic stresses and residual stresses.Induced stresses are man-made stress component due to removal or addition ofmaterial. 2
- 3. Ground Control ???No excavation No Problem All rock in the ground subjected to compressive stresses and if excavation is made the rock left standing has to take more load because the original support provided by the rock within the excavation has been removed. Effect of vertical force line on u/g opening• An underground opening brings about a lateral squeeze of vertical force lines. This results in a large stress concentration at the abutments a and b. 3
- 4. Induced stresses• It is man-made stress component due to removal or addition of material.• The reasons for induced stress conditions in rock are: – Loosening of the rock mass around cavities; – Weight of the overburden rock mass; – Tectonic forces, and – Volumetric expansion of the rock mass by thermal effects, or by swelling brought about by the action of physical or physico- chemical processes.• Depending upon the kind and properties of the rock, induced stress conditions require the use of different construction methods. All of the above-mentioned four induced stress (pressure) conditions may also occur simultaneously. Division of area around an excavation• Near Field – Adjacent to the excavations – Area of interest to the designer• Far Field – Remote from excavation – The response of the rock is essentially elastic 4
- 5. Requirements for Stress Analysis?? – Rock Mass Property – Geological Discontinuities • Fracture patterns, Fault zones, Joints, Bedding planes etc – Excavation Geometry Stress Assumptions• Prior to disturbance of rock mass, it is visualized that the rock is subjected to the effect of the rock-forming forces as well as the force of gravity.• Vertical stresses are equal to σv = ρgh• The horizontal stresses are generated by assuming that the strain is zero.σh = (ν/1-ν)σv =k σvFor example: ν = 0.3; σh = 0.43 σv 5
- 6. Vertical and Horizontal stresses Vertical Stress (after Brown Townend and Zoback, (2000) and Hoek, 1978) Ratio of Horizontal to Vertical Stress Sheory,1994 1 K = 0.25 + 7 Ek 0.001 + zwhere Ek (GPa) is the average deformation modulus of the upper part of theearth’s crust measured in a horizontal direction. 6
- 7. Stresses around mine openings• The underground rock structures (openings) are the excavations which are created in a pre- stressed environment. Stress analysis provide insight into the changes in preexisting stress equilibrium caused by an opening. It interprets the performance of an opening in terms of stress concentrations and associated deformations and serves as a rational basis for establishing the performance of requirements for design.• The properties of the rockmass are complex, and no single theory is available to explain rockmass behaviour. However, the theories of elasticity and plasticity provide results that have relevance to the stress distributions induced about openings and provide a first step to estimating the distribution of stresses around openings.• Prior to excavation, the in situ stresses in the rockmass are in equilibrium. Once the excavation is made, the stresses in the vicinity of the opening are redistributed and stress concentrations develop. 7
- 8. Design Approach• Analytical solution• Empirical Solution• Physical Modeling• Numerical Modeling Various cross section of Underground openings Circular Analytical solution by Kirsch in 1898. Elliptical Analytical solution by Bray in 1976. Parabolic cross-section Square Horse- Shoe Shape Rectangular 8
- 9. Circular Opening• Prediction of the stresses and displacements around a circular opening in the rock mass at great depth is an important problem in geotechnical, petroleum and mining engineering such as the design of tunnels, boreholes and mine shafts. Stress concentration around mine openings 9
- 10. Kirsch Equation (1898)• Analytical or Closed form solution for circular opening in 2D.• Continuous, Homogenous, Isotropic and Linear Elastic material (CHILE).• Circular opening of radius, a• In Polar co-ordinate system find the radial, tangential and shear stress as well as radial and tangential displacement at a point located at (r,θ).Krisch’s equations for stresses around mine openings 1 a2 a2 a4 σ r = σ z (1 + k ) 1 − 2 − (1 − k ) 1 − 4 2 + 3 4 cos 2θ 2 r r r 1 a2 a4 σ θ = σ z (1 + k ) 1 + 2 + (1 − k ) 1 + 3 4 cos 2θ 2 r r 1 a2 a4 τ rθ = σ z (1 − k ) 1 + 2 2 − 3 4 sin 2θ 2 r r 10
- 11. Inferences from kirsch equationFor any excavation shape, only tangential stresses exist at the boundary of an excavation (without internal loading), for example: put a/r = 1For, r = 4a, the ratio of induced to applied stress are close to unity, means no influence of opening beyond that.For a hydrostatic stress field (k =1), – The stress concentration is 2 times SV or SH everywhere on the boundary. – No, shear stress any where within rock mass.For k = 0, maximum stress concentration is 3 (i.e. compression) and minimum stress concentration is -1(Tensile).Stress is independent of elastic constant like young’s modulus and poisons ratio.The equation is not including the radius, but a ratio a/r (dimensionless) (dimensionless) i.e. the stress at boundary of an excavation are independent of absolute value of the radius. 11
- 12. Fig: Stress concentration along the wall and roof of a rectangular opening The stress concentrations along the perimeter of anopening show a large increase for non-circular openings orangular openings.Hoek & Brown -determined tangential stresses on the excavation surface at the crown .and in the sidewall for different –shaped openings for a range of in situ stress ratios 12
- 13. Importance of Elastic analysis• Maximum and minimum stresses on the boundary of the opening• Boundary displacement induces by the excavation• Extent of zone of influence• The extent of the overstresses region• The increase in strain energy, and the dynamic energy released, when an excavation is generated Extent of Plastic Zone 13
- 14. Zone of Disturbance Upon blasting out an opening, the rock becomes disintegrated. Loose zone, or zone of disturbance between the contour of the opening and the plastic zone, is created. Because of the disintegrated condition of the rock, the rock here is not in a plastic state. It does not have any unconfined compressive strength, nor can it take and carry any stress.Elastic and plastic zones around circular opening 14
- 15. Design methods for single opening• Any opening will be stable if the maximum stress occurring around the opening is less than the failure strength of the rocks defined in a failure criterion.• Simplest method of designing is to determine what type of opening in what geometry produces the maximum stress and compare it with the failure strength of the rocks. Stress Shadow • First case shows stresses amplified between excavations. • Second case shows stresses attenuation. 15
- 16. Zone of Influence • The zone around an excavation in which the stresses are perturbed from their in-situ values by more than a defined amount. • For example, we could define the zone of influence around the excavation as the zone within which at least one component of the stress tensor is perturbed by greater than, say, 5% of its in situ value, expressed mathematically as σ Induced − σ Natural ≥ 0.05σ Natural r5% ≤ a 20 Excavation sequencing alternatives Should we create excavation I or excavation II first ?• Advantage of creating smaller dia opening II first is that the final stress field acting on Excavation I will be in place before that excavation is made, and the process of creating Excavation I will not appreciably affect Excavation II.• Advantage of creating large dia Excavation I first is that the excavation is made in an undisturbed stress field, and the tunnel can be supported in anticipation of the stresses that will be induced following the creation of Excavation II. 16
- 17. Conclusion• For the analysis of stability of underground openings, the knowledge of stresses, strength and failure mechanism are important. The idea of the stress concentrations and their effects on the surroundings of the openings helps the design engineers to plan a suitable method of support system. However the knowledge of rockmass properties are still to be acquired and rockmass classification systems is an attempt towards the purpose. 17

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